A271663 Convolution of nonzero squares (A000290) with nonzero pentagonal numbers (A000326).

1, 9, 41, 131, 336, 742, 1470, 2682, 4587, 7447, 11583, 17381, 25298, 35868, 49708, 67524, 90117, 118389, 153349, 196119, 247940, 310178, 384330, 472030, 575055, 695331, 834939, 996121, 1181286, 1393016, 1634072, 1907400, 2216137, 2563617, 2953377, 3389163, 3874936

Offset

0,2

Comments

More generally, the ordinary generating function for the convolution of nonzero h-gonal numbers and k-gonal numbers is (1 + (h - 3)*x)*(1 + (k - 3)*x)/(1 - x)^6.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

OEIS Wiki, Figurate numbers

Eric Weisstein's World of Mathematics, Square Number

Eric Weisstein's World of Mathematics, Pentagonal Number

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Formula

O.g.f.: (1 + x)*(1 + 2*x)/(1 - x)^6.

E.g.f.: (120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*x^5)*exp(x)/120.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

a(n) = (n + 1)*(n + 2)*(n + 3)*(6*n^2 + 19*n + 20)/120.

Sum_{n>=0} 1/a(n) = 1.149165731...

Mathematica

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 9, 41, 131, 336, 742}, 40]
Table[(n + 1) (n + 2) (n + 3) (6 n^2 + 19 n + 20)/120, {n, 0, 40}]
With[{nmax = 50}, CoefficientList[Series[(120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*x^5)*Exp[x]/120, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 07 2017 *)

Prog

(PARI) vector(40, n, n--; (n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120) \\ Altug Alkan, Apr 12 2016
(Magma) /* From definition: */ P:=func<n, k | (n^2*(k-2)-n*(k-4))/2>; /*, where P(n, k) is the n-th k-gonal number, */ [&+[P(n+1-i, 4)*P(i, 5): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 12 2016
(Magma) [(n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120: n in [0..40]]; // Bruno Berselli, Apr 12 2016

Crossrefs

Cf. A000290, A000326.

Cf. A005585: convolution of nonzero squares with nonzero triangular numbers.

Cf. A033455: convolution of nonzero squares with themselves.

Cf. A051836 (after 0): convolution of nonzero triangular numbers with nonzero pentagonal numbers.

Sequence in context: A297740 A297741 A001846 * A034441 A201275 A056243

Adjacent sequences: A271660 A271661 A271662 * A271664 A271665 A271666

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Apr 12 2016

Extensions

Edited by Bruno Berselli, Apr 12 2016

Status

approved